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arXiv · 2204.02689

Existence of a Non-Zero $(0,1)$-Vector in the Row Space of Adjacency Matrices of Simple Graphs

Sudip Bera
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Abstract

We look for a non-zero $(0, 1)$-vector in the row space of the adjacency matrix $A(Γ)$ of a graph $Γ,$ provided $Γ$ has at least one edge. Akbari, Cameron, and Khosrovshahi conjectured that there exists a non-zero $(0,1)$-vector in the row space of $A(Γ)$ (over the real numbers) which does not occur as a row of $A(Γ).$ This conjecture can be easily verified for graphs having diameter is equal to $1$ (complete graphs). In this article, we affirmatively prove this conjecture for any graph whose diameter is $\geq 4.$ Furthermore, in the remaining two cases that is, for graphs with diameter is equal to $2$ or $3,$ we report some progress in support of the conjecture.

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Sudip Bera. 2025-02-25. Existence of a Non-Zero $(0,1)$-Vector in the Row Space of Adjacency Matrices of Simple Graphs. https://doi.org/10.1007/s40840-025-01838-0

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